Rigid-Recurrent Sequences for Actions of Finite Exponent Groups

TL;DR

This paper explores the construction of special sets in finite cyclic groups to understand the interplay of rigidity, weak mixing, and recurrence in dynamical systems, extending previous results to all q > 1.

Contribution

It generalizes the existence of Kronecker-type sets generating dense subgroups from prime q to all q > 1, advancing the understanding of rigidity sequences in abelian group actions.

Findings

Constructed Kronecker-type sets for all q > 1.

Established the existence of large rigidity sequences for weak mixing systems.

Extended prior results from prime q to all q > 1.

Abstract

The focus of this paper is to better understand the coexistence of rigidity, weak mixing, and recurrence by constructing thin sets in the product of countably many copies of the finite cyclic group of order q. A Kronecker-type set K is a subset of this group on which every continuous function into the complex unit circle equals the restriction, to K, of a character in the group's Pontryagin dual. Ackelsberg proves that if, for all q > 1, there exists a perfect Kronecker-type set generating a dense subgroup, then there exist large rigidity sequences for weak mixing systems of actions by countable discrete abelian groups. Ackelsberg shows the existence of such sets for prime values of q, while we construct them for all q > 1.